356 research outputs found
Algebraic entropy for differential-delay equations
We extend the definition of algebraic entropy to a class of
differential-delay equations. The vanishing of the entropy, as a structural
property of an equation, signals its integrability. We suggest a simple way to
produce differential-delay equations with vanishing entropy from known
integrable differential-difference equations
On the algebraic structure of rational discrete dynamical systems
We show how singularities shape the evolution of rational discrete dynamical
systems. The stabilisation of the form of the iterates suggests a description
providing among other things generalised Hirota form, exact evaluation of the
algebraic entropy as well as remarkable polynomial factorisation properties. We
illustrate the phenomenon explicitly with examples covering a wide range of
models
Scenarios to explain extreme Be depletion in solar-like stars: accretion or rotation effects ?
Studies of beryllium abundance in large samples of solar-type stars show a
small fraction of extremely beryllium-deficient stars, which challenges our
current understanding of light element depletion in these stars. We suggest two
possible scenarios that may explain this high level of Be depletion: early
accretion and rotational mixing. We show that in both cases, the conditions
required to reach the observed level of Be depletion are quite extreme, which
explains the very small fraction of detected Be outliers. We suggest that
substantial Be depletion can be obtained in stars if they were fast rotators in
the past, with high initial rotational velocities and short disc lifetimes. Our
analysis suggests that rotational mixing may not be efficient enough to deplete
Be in less than 10 Myr. Consequently, the detection of strongly Be-deficient
stars in clusters younger than 10 Myr may provide a genuine signature of
accretion process and the proof that some protostars may undergo many extreme
bursts of accretion during their embedded phases of evolution.Comment: 7 pages, 6 figures, accepted for publication in A&
An exercise in experimental mathematics: calculation of the algebraic entropy of a map
We illustrate the use of the notion of derived recurrences introduced earlier
to evaluate the algebraic entropy of self-maps of projective spaces. We in
particular give an example, where a complete proof is still awaited, but where
different approaches are in such perfect agreement that we can trust we get to
an exact result. This is an instructive example of experimental mathematics
On the relevance of bubbles and potential flows for stellar convection
Recently Pasetto et al. have proposed a new method to derive a convection
theory appropriate for the implementation in stellar evolution codes. Their
approach is based on the simple physical picture of spherical bubbles moving
within a potential flow in dynamically unstable regions, and a detailed
computation of the bubble dynamics. Based on this approach the authors derive a
new theory of convection which is claimed to be parameter free, non-local and
time-dependent. This is a very strong claim, as such a theory is the holy grail
of stellar physics.
Unfortunately we have identified several distinct problems in the derivation
which ultimately render their theory inapplicable to any physical regime. In
addition we show that the framework of spherical bubbles in potential flows is
unable to capture the essence of stellar convection, even when equations are
derived correctly.Comment: 14 pages, 3 figures. Accepted for publication in Monthly Notices of
the Royal Astronomical Society. (Comments and criticism are welcomed
Lithium depletion in solar-like stars: effect of overshooting based on realistic multi-dimensional simulations
We study lithium depletion in low-mass and solar-like stars as a function of
time, using a new diffusion coefficient describing extra-mixing taking place at
the bottom of a convective envelope. This new form is motivated by
multi-dimensional fully compressible, time implicit hydrodynamic simulations
performed with the MUSIC code. Intermittent convective mixing at the convective
boundary in a star can be modeled using extreme value theory, a statistical
analysis frequently used for finance, meteorology, and environmental science.
In this letter, we implement this statistical diffusion coefficient in a
one-dimensional stellar evolution code, using parameters calibrated from
multi-dimensional hydrodynamic simulations of a young low-mass star. We propose
a new scenario that can explain observations of the surface abundance of
lithium in the Sun and in clusters covering a wide range of ages, from
50 Myr to 4 Gyr. Because it relies on our physical model of convective
penetration, this scenario has a limited number of assumptions. It can explain
the observed trend between rotation and depletion, based on a single additional
assumption, namely that rotation affects the mixing efficiency at the
convective boundary. We suggest the existence of a threshold in stellar
rotation rate above which rotation strongly prevents the vertical penetration
of plumes and below which rotation has small effects. In addition to providing
a possible explanation for the long standing problem of lithium depletion in
pre-main sequence and main sequence stars, the strength of our scenario is that
its basic assumptions can be tested by future hydrodynamic simulations.Comment: 7 pages, 3 figures, Accepted for publication in ApJ Letter
Solvable Chaos
We present classes of discrete reversible systems which are at the same time chaotic and solvable
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